Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass

Abstract

This work aims to present an analytical study on the dynamics of a third body in the restricted three-body problem. We study this model in the context of the third body having variable-mass changes according to Jeans' law. The equation of motion is constructed when the variation of the mass is non-isotropic. We find an appropriate approximation for the locations of the out-of-plane equilibrium points in the special case of a non-isotropic variation of the mass. Moreover, some graphical investigations are shown for the effects of the parameters which characterize the variable mass on the locations of the out-of-plane equilibrium points, the regions of possible and forbidden motions of the third body. This model has many applications, especially in the dynamics behavior of small objects such as cosmic dust and grains. It also has interesting applications for artificial satellites, future space colonization or even vehicles and spacecraft parking.Keywords Restricted three-body problem · Variable mass · Out of plane equilibrium points B E.I. Abouelmagd

On the libration collinear points in the restricted three – body problem

“…For instance, the existence of libration points, their stability and the periodic orbits in the proximity of these points under the oblateness, triaxialty of the primaries or the effect of photogravitational force or combination of them are studied by Sharma [3], Singh and Ishwar [4], Sharma et al [5,6], Singh and Mohammed [7], Abouelmagd and El-Shaboury [8], Abouelmagd [9], Abouelmagd [10,11], Abouelmagd et al [12], Abouelmagd and Sharaf [13], Abouelmagd et al [14,15], Abouelmagd et al [1,16,17,18], Abouelmagd and Mostafa [19], Abouelmagd et al [20], Abouelmagd and Guirao [21].…”

Stability of the equilibrium points in the circular restricted four body problem with oblate primary and variable mass

“…We shall follow the procedure of Abouelmagd [1]. The equations of motion for the infinitesimal variable mass when the variation of mass is non-isotropic and originating from one point are…”

“…Using the relation between the inertial and rotating coordinates (Abouelmagd [1]), the equations of motion in a rotating coordinate system for an infinitesimal variable mass P will be…”

“…Researchers have also studied about restricted three and four body problem with variable mass ie. Shrivastava [13], Singh [14], [15], [16], [17], Zhang [19], Abouelmagd [1]. After reviewing all the literature, we impressed to do the work on the restricted four body problem with one of the primaries as oblate body and all the three primaries are placed at the vertices of a triangle and also the infinitesimal mass is taken as variable mass.…”

The effect of perturbations on the circular restricted four-body problem with variable masses

“…They also illustrated the periodic orbits, the Poincare surface of section and their stability under the effect of radiation pressure. Abouelmagd in [3][4][5][6] explored the restricted three-body problem with different perturbations. Papadakis in [40,41] studied the 3D symmetric periodic orbits of the circular restricted four-body problem, through their bifurcation from the plane orbits.…”